Program gives the code for the `LLRotation`
procedure of the `AVLTree` class.
This code implements the LL rotation shown in Figure .
The purpose of the `LLRotation` member function
is to perform an LL rotation at the root of a given AVL tree instance.

**Program:** `AVLTree` Class `LLRotation` Member Function Definition

The rotation is simply a sequence of pointer manipulations
followed by two height adjustments.
Notice the rotation is done in such a way so that the
the given `AVLTree` instance remains the root of the tree.
This is done so that if the tree has a parent,
it is not necessary to modify the contents of the parent.

The `AVLTree` class also requires an `RRRotation` member
function to implement an RR rotation.
The implementation of that function
follows directly from Program .
Clearly, the running time for the single rotations is *O*(1).

Program gives the implementation for the
`LRRotation` member function of the `AVLTree` class.
This double rotation is trivially implemented as a sequence of
two single rotations.
As above, the routine for the complementary rotation
is easily derived from the given code.
The running time for each of the double rotation functions is also *O*(1).

**Program:** `AVLTree` Class `LRRotation` Member Function Definition

When an imbalance is detected, it is necessary to correct the imbalance
by doing the appropriate rotation.
The code given in Program takes care of this.
The `Balance` routine tests for an imbalance by calling
the `BalanceFactor` function.
The balance test itself takes constant time.
If the node is balanced, only a constant-time height adjustment is needed.

**Program:** `AVLTree` Class `Balance` Member Function Definition

Otherwise, the `Balance` routine of the `AVLTree` class
determines which of the four cases has occurred,
and invokes the appropriate rotation to correct the imbalance.
To determine which case has occurred,
the `Balance` routine calls the `BalanceFactor`
function two more times.
Therefore, the time for selecting the case is constant.
In all only one rotation is done to correct the imbalance.
Therefore, the running time of this routine is *O*(1).

The `Insert` routine for AVL trees is inherited from
the `BST` class (see Program ).
The very last thing that routine does is to call the `Balance` function.
which has been overridden.
As a result the `Insert` routine
adjusts the heights of the nodes along the insertion path
and does a rotation when an imbalance is detected.
Since the height of an AVL tree is guaranteed to be ,
the time for insertion is simply .

Copyright © 1997 by Bruno R. Preiss, P.Eng. All rights reserved.